My research program aims to uncover the full scope of dynamics in general relativity. In many cases, perturbations of stationary background spacetimes quickly damp away due to propagation of gravitational waves to infinity or through a horizon. However, these dissipation channels are sometimes absent or weak, resulting in slow decay or even unstable growth. When this occurs, nonlinear gravitational dynamics take hold and determine the final state. Examples include the superradiant instability, the instability of anti-de Sitter (AdS) spacetime, and the equilibration of large AdS black holes. These cases display nontrivial behavior and even turbulence of the spacetime geometry. This research involves a variety of numerical and analytic approaches, including the canonical energy method and multiscale analysis. A major finding is the emergence of new approximate conservation laws that control the turbulent cascades. I conclude with a discussion of my future plans to adapt these methods to new systems of mathematical and astrophysical importance.